Question

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5$ are the roots of the equation $$ x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 = 0, $$ then find the value of $$ \frac{1}{\alpha_1^2} + \frac{1}{\alpha_2^2} + \frac{1}{\alpha_3^2} + \frac{1}{\alpha_4^2} + \frac{1}{\alpha_5^2}. $$

• A. \( 15 \)
• B. \( \frac{1}{7} \)
• C. \( 7 \)
• D. \( 12 \)

Hint:

For symmetric polynomials involving reciprocals, use Vieta’s formulas to express sums in terms of the polynomial’s coefficients.

Answer 1

The Correct Option is C

Step 1: Using the identity for sum of reciprocals of squares
We use the identity: \[ \sum_{i=1}^{5} \frac{1}{\alpha_i^2} = \left( \sum_{i=1}^{5} \alpha_i^2 \right) - 2 \sum_{1 \leq i<j \leq 5} \alpha_i \alpha_j. \] Step 2: Finding required symmetric sums
From Vieta’s formulas applied to the polynomial equation: \[ x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 = 0, \] we obtain the sums: \[ \sum \alpha_i = 5, \quad \sum \alpha_i \alpha_j = 9, \quad \sum \alpha_i \alpha_j \alpha_k = 9, \quad \sum \alpha_i \alpha_j \alpha_k \alpha_l = 5, \quad \prod \alpha_i = 1. \] Using the square identity: \[ \sum \alpha_i^2 = (\sum \alpha_i)^2 - 2 \sum \alpha_i \alpha_j = 5^2 - 2(9) = 25 - 18 = 7. \] Thus: \[ \sum_{i=1}^{5} \frac{1}{\alpha_i^2} = 7. \] Step 3: Conclusion
Hence, the final answer is: \[ \boxed{7}. \]